Quantum Mechanics in 3D with the PDE Toolbox

Computes eigenstates of the Schrodinger equation in three dimensions for various geometries and potential functions.

This educational Live Script explores three-dimensional stationary state solutions to the Schrodinger equation with the Partial Differential Equation Toolbox, extending the one-dimensional, two-dimensional, and three-dimensional examples in Ref[1-6]. The script illustrates Toolbox support for the creation and meshing of complex three-dimensional geometries. The numerical results for the 3-D harmonic oscillator and Coulomb potentials (both possibly anisotropic) are compared to analytic results. The case of an isotropic spherical harmonic oscillator confined between two spherical shells is solved by finding the solutions to the radial equation numerically and the energy spectrum compared to the full 3-D calculation as a further check.

This script may interest students and instructors in physics and related fields. 'Try this' suggestions, 'Challenges,' hyperlinks, and references are included for further exploration.